Engineering Optimization: Theory and Practice, Fourth Edition

7.13 Interior Penalty Function Method 437

effective in reducing the disparities between the contributions of the various constraints
to theφfunction.

Example 7.7

Minimizef (x 1 , x 2 )=^13 (x 1 + 1 )^3 +x 2

subject to
g 1 (x 1 , x 2 ) =−x 1 + 1 ≤ 0
g 2 (x 1 , x 2 ) =−x 2 ≤ 0

SOLUTION To illustrate the interior penalty function method, we use the calculus
method for solving the unconstrained minimization problem in this case. Hence there
is no need to have an initial feasible pointX 1. The φfunction is

φ(X, r)=

1

3

(x 1 + 1 )^3 +x 2 −r

(

1

−x 1 + 1

−

1

x 2

)

To find the unconstrained minimum ofφ, we use the necessary conditions:

∂φ

∂x 1

= (x 1 + 1 )^2 −

r

( 1 −x 1 )^2

= 0 , that is, (x^21 − 1 )^2 =r

∂φ

∂x 2

= 1 −

r

x 22

= 0 , that is, x^22 =r

These equations give

x 1 ∗ (r)=(r^1 /^2 + 1 )^1 /^2 , x 2 ∗(r)=r^1 /^2

φmin(r)=^13 [ r(^1 /^2 + 1 )^1 /^2 + ] 13 + 2 r^1 /^2 −( 1 /r)−( 1 /r 31 / (^2) + 1 /r (^2) ) 1 / 2
To obtain the solution of the original problem, we know that
fmin= iml
r→ 0
φmin(r)
x 1 ∗= iml
r→ 0
x 1 ∗(r)
x 2 ∗= iml
r→ 0
x 2 ∗(r)
The values off,x∗ 1 , andx 2 ∗corresponding to a decreasing sequence of values ofrare
shown in Table 7.3.
Example 7.8
Minimizef (X)=x 13 − 6 x^21 + 11 x 1 +x 3
subject to
x 12 +x^22 −x 32 ≤ 0
4 −x 12 −x^22 −x 32 ≤ 0
x 3 − 5 ≤ 0
−xi≤ 0 , i= 1 , 2 , 3